From microscopic kinetics to generalized allen-cahn equations. Application to adatoms and intercalation dynamics

Abstract

A mean-field approximation of stochastic lattice gas models allows to write generalized Cahn-Hilliard and Allen-Cahn equations. The Mean Field Kinetic Equations (MFKE) obtained in this way have been used recently to perform analytic calculations and numerical simulations of various physical phenomena, like dendritic growth, spinodal decomposition of droplets and films, or transport properties in presence of ordered domains [Gouyet, 1995, Plapp and Gouyet, 1997a,b, Nassif et al., 1998]. In the present conference we show how such MFKE with repulsive interacting species can be used to understand ions intercalation dynamics in certain kinds of host lattices, and the particular diffusion behaviour of interacting adatoms on a crystalline surface. For that purpose we consider stochastic square or simple cubic lattice gases with nearest neighbour interactions. In both lattices phase ordering leads to a symmetry breaking into two sublattices, i.e. a checkerboard structure in the case of the square lattice. The MFKE then consist in a set of two coupled equations, one for the kinetics of the mean concentration (a conserved quantity), a second for the kinetics of the relative occupation between the two ”black and white” checkerboard-like sublattices (non conserved order parameter). Introduction of simultaneously evolving sublattices allows to write the mean-field equations in a compact simple form. Solutions of these equations show various interesting properties: 1) the diffusion coefficient decreases in the ordered regions. In first approximation it is possible to define an effective diffusion coefficient which only depends on the order parameter calculated at equilibrium. However we observe an incorrect behaviour at the critical concentrations, and in addition, 2) ordering is not abrupt and appears outside the concentration range of the equilibrium phase diagram. This leads to consider kinetic phase diagrams which take into account finite concentration gradient and curvature effects. We show that in the vicinity of the critical concentrations the transition line is smooth and we determine its scaling as function of time. 3) We also find a change in the mean concentration inside the antiphase boundaries. We have determine its characteristics and calculated the associated increase of the diffusion coefficient in these interfaces. The above results apply to recent experiments of diffusion of Pb adatoms on (100) copper surface [Cohen et al. 1993], which present ordered domains and an important modification of the diffusion coefficient, measured by Rutherford backscattering. It should also allow to interpret voltametric measurements during the intercalation of lithium in LiTiS₂ [Chabre and Deniard, 1987], in which again ordering modifies strongly the intercalation behaviour.